Question: Kreps and Wilson (1982) consider the following war of attrition. There are two players, i = 1, 2. Time is continuous from 0 to 1. When one player concedes, the game ends. Each player can be either "strong" (with probability p for player 1 and q for player 2) or "weak" (with probabilities 1 p and I - q). A strong player enjoys fighting and therefore never concedes. A weak player 1 (respectively, a weak player 2) loses 1 per unit of time while fighting and makes a >0 (respectively, b>0) per unit of time when his rival has conceded. Thus, a weak player 1 has payoff a(1 - 0-1 when it wins at t and payoff -t when it concedes at t. There is no discounting.
(a) Show that from time 0+ on, the posterior beliefs p, and q, of each player about the other must belong to the curve q = pb/a.
(b) Show that one of the weak types exits with positive probability at date 0 exactly (that is, a player's cumulative probability distribution of exit times exhibits an atom at t = 0). How are the weak types' payoffs affected by a, h. p. and q?